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A complete generalization of Göllnitz’s “big” theorem

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Abstract

In the spirit of Göllnitz’s “big” partition theorem of 1967, we present a new mod-6 partition identity. Alladi et al. provided a four-parameter refinement of Göllnitz’s big theorem in 1995 via a key identity of generating functions and the method of weighted words. By means of this technique, two similar mod-6 identities of this type were discovered—one by Alladi in 1999 and one by Alladi and Andrews in 2015. We finish the picture by presenting and proving the fourth and final possible mod-6 identity in this spirit. Furthermore, we provide a complete generalization of mod-n identities of this type. Finally, we apply a similar argument to generalize an identity of Alladi et al. from 2003.

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Acknowledgements

All the results in this paper were proved in June and July, 2016 (except for the proof of Theorem 6.3, which was fixed in August, 2017) in an REU project at Rutgers University with T.C. and J.K. as the participants and M.C.R. as research mentor. We would like to thank James Lepowsky for making this research project possible—for creating this REU project and putting its members together, brainstorming ideas for research directions, and providing guidance. This project was completed through the 2016 DIMACS (Center for Discrete Mathematics and Theoretical Computer Science) REU program with funding from the Rutgers Mathematics Department and the National Science Foundation.

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  1. Jongwon Kim

    Present address: Department of Mathematics, University of Pennsylvania, Philadelphia, PA, 19104, USA

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  1. Department of Mathematics, Rutgers University, Piscataway, NJ, 08854, USA

    Terence Coelho, Jongwon Kim & Matthew C. Russell

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  1. Terence Coelho
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  2. Jongwon Kim
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Correspondence to Terence Coelho.

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Coelho, T., Kim, J. & Russell, M.C. A complete generalization of Göllnitz’s “big” theorem. Ramanujan J 55, 73–102 (2021). https://doi.org/10.1007/s11139-020-00262-1

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  • DOI: https://doi.org/10.1007/s11139-020-00262-1

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